Thank you. I was explaining to my friend why in "order of operations" that multiplication and division were interchangeable same with addition and subtraction. And when i said "because basically theyre the same thing" she looked at me as if i was crazy 😂
@@savazeroa try solving 5+2*6 solely going left to right without using order of operations. That's how many eggs I collected from the chooks over the past 3 days, so there is a correct answer: 17
@@SbF6H the thing with these equations is that, if you dont know what it represents, its very difficult to reverse engineer what it represents even if you know the notation unlike some simpler equations. i personally didnt know it but its pretty easy to understand.
A future original 3B1B in the making, keep up the great work and amazing videos. Would love to see longer videos if it meant minimizing holes and gaps. Thank you for your work!
@@Shadoxite from my other commentary (only watched a few minutes and stopped): "Tetration is useful in combinatorics and pure mathematics is not about applications. Applications are easy and they will always show up sooner or later. Root and 1/p power are not the same operations. Only if you match the domains specifically and pick a certain root, you could somewhat bring them together say for R+. They are two very different beasts if you properly consider complex numbers. Why are you confusing people if you don't understand the basics yourself?"
I am so glad that Im getting an explanation for complex math, my teachers all basically taught me was "this is how you use them because this is how they are used" but now why they are used that way or what they are, and for me to learn I have to actually understand something at a fundamental level
Got to hand it to you mate, although i knew these concepts beforehand, the visualization and most importantly your explanations were amazing, very underrated video, amazingly put
I always find it so odd that people struggle so much with algebra. Probably a result of it being taught way too late, as substitution is so basic that it really should just be taught around the same time as multiplication (and should be followed within a year or two by parentheses and factorisation, as they're another one people tend to struggle with due to how late they're taught.)
I see people struggling with Fractions, it's so easy, it's literally just division and people struggle with it, in my opinion they should only teach fractions and avoid pure division as much as possible, because in the future(High school) these people won't use "÷" anymore and will only use fraction.
@@reclaimer2019 you could probably teach ÷ when teaching other alternative notations like *, ^, and ↑↑↑ and just teach them like you would alternative characters in English like @, &, etc. Though you can always just teach both division and fraction notation simultaneously as different was of writing it, as ÷ is really important for factorisation, as 1/x(2+3) [2*(1/x)+3*(1/x)] and 1÷x(2+3) [1÷(2x+3x)] aren't the same thing [x=1, 1/1(2+3)=5, 1÷(2*1+3*1)=1/5]. You Can get around this with 1/(x(2+3)) or a long fraction sign that I don't feel like looking for the unicode for, but a division sign does the job just fine too.
@@reclaimer2019they should be taught that these are equal, also, the notation for a single line equation can get very messy, but it makes absurd sense. Like how 1/1+1 is different than 1/(1+1), but some people seem to not be able to recognize this.
People struggle with algebra due to the fact it makes no sense. This is because algebra in Western countries isn't taught systematically but with an adhoc approach. When we were going over equations we never went over what operations you can do to them. Also parentheses aren't explained well usually. For example something like this 5+(5-4) would be "incorrect" to solve as 5+5-4=6 even though the parenthesis in this case do nothing.
I love why math works and I’m glad more people are covering it in depth. You should do mechanics next, it’s pretty easy to explain how we get the laws of motion and why things like energy are useful
Not exactly sure why I watched the entire video, considering I've done all that in depth throughout my academic journey, but damn, that's an easy to grasp and extremely quick explanation to lots of interchanging mathematical concepts that I was taught through years of math classes. Honestly well done. Had this existed half a decade ago, it would have made my life way more "understandable" (definitely not easier - applying everything mentioned here to actual use is why proper education takes years, not 30 minutes).
Quick note at 16:00, dy/dx is actually the derivative f’(x) Whereas if we want to do the action of taking the derivative of f(x), We gotta write out d/dx f(x). Think of d/dx as the derivative operator, Just like how x tells us to multiply, d/dx tells us to take the derivative While dy/dx = f’(x)
30:43 thank you man. i feel so validated. i tried explaining to everyone i could that sines and cosines just don't feel usable. un-graspable and undefined. but here they are. in their true form. beautiful.
another way to write sin and cos: sin(z) = (e^(iz) - e^(-iz))/(2i) cos(z) = (e^(iz) + e^(-iz))/2 This format makes them easier to use with complex inputs z, can help you prove derivative and integral trig properties, as well as shows the connection to the hyperbolic trig functions sinh and cosh.
This didn't sit right with me and i kept mentioning it during the stream this was being made I personally would define sin and cos by their infinite taylor series, of course, the formula for the taylor series requires the derivatives of sin and cos respectively, but in the case of sin and cos they're nice infinite sums (for the maclauren series) technically, i think maybe this is a circular definition as the motivation behind taylor series involves the derivatives of sin and cos, and we're using that to define sin and cos, but i can't think of anything better- Defining them in terms of complex exponentiation would require a definition of complex exponentiation If you define complex exponentiation by plugging i into the taylor series of e^x, and then proving e^ix is equal to cos(x)+isin(x), (using the taylor series of cos(x) and sin(x)) you're still using the taylor series. if you don't want to use the taylor series, and just define complex exponentiation by euler's formula, you still have cos and sin in eulers formula! it's a circular definition! Please tell me where i'm wrong- i think i'm probably wrong
@@savazeroa @savazeroa no you're 100% correct, i noticed that in the vid as well that it seemed self-referntial and kinda reduntant but i guess he didn't wanna go on a tangent to explain series but yh defining them with their series expansion would be more correct than what is shown
The derivate of sin and cosin osciolating but never changing property now makes so much sense considering theyre made up of exponetionals. Thanks for revealing some deep math knowledge that will help me in my Electrical escapades.
I have been struggling with my digital signals and systems course because I was afraid of notation, and I did not completely understand the transition from complex numbers and euler's formula into the Fourier Transform. It's the day before my second midterm, and this video might help me save my grade. Thank you so much, and please make more videos like this to help us engineering students!
As a maths enjoyer, I have no Idea what a normal person would think watching this... But for me, I absolutely love this content! You display it very well.
This is AMAZING. Thank you for making it. I've just finished an AP math course (basic 1st year math in hs ) and this went through and beyond all my knowledge 😅
@@Snakehandler268 something funny i understood everything in this video and only watched it twice im in 8th grade and my teachers face when i started explaining Fourier transform
Perhaps. But my opinion is that people just aren’t required to, or simply don’t, do enough problems on their own. Math isn’t something you can cram for just before an exam, like, say, history. You need to develop an intrinsic understanding that practically always takes a lot of time, effort and practice.
@chudleyflusher7132 This is true and if you stop using it after high school you will forget things quickly. I had to relearn algebra from the ground up 2 years after I graduated high school cause I took a gap year and solved zero math.
That was honestly the most well explained video on maths I have ever seen. My friends often struggle to understand why I find maths so exciting, but I'm pretty sure they'll understand once they watch this. I loved the flow of introducing all of the topics, as well as the animations which made it super easy to not get lost in all of the new words for someone whose first language isn't English. Thank you so much for this wonderful piece of media contributed to the internet, I'll make sure to recommend your channel to as many people as I can.
This is math from the 19th century at most, so very much not all of math. The things people are currently doing in math is a lot lot lot more complicated.
It was so jarring to be introduced to math like "Divide rise by run DO IT DO IT NOW OR YOU GET BAD GRADE" instead of explaining it more organically like this.
Hi Matik! I've just begun trigonometric, it's hard to approach at the begin, but your video makes me more certainty about mathematic.Great video and greetings from Italy!!!
Top 5 ytbers imo, and remember, aside from Vsauce, this is the only guy that does anything academic in the top 5 I think you have no idea how good ur vids are. Now if you did this with physics THEN I actually straight up explode
I've messed with all of these functions and haven't felt like I've ever had a better understanding then right now after watching this video. I'm sure the average person will need more so please keep up the incredible work that you're doing!
Excellent video. For the sake of simplicity, we're often taught these topics without any further explanation as to how they were derived and where they were derived from which can leave one with a lot of questions. This video does a great job laying it all out with fun graphics. Subbed!
24:57 but the term -1² doesnt equal to 1 it equals to -1 because first you square -1 and then multiply it by negative, you should have writed (-1)² which equals to 1
This was… incredible!! I absolutely love your videos and how you build up concepts. Your visuals are spectacular and your explanations show an amazing and unique ability to communicate concepts in a way that is absolutely perfect for anyone who just feels like “they don’t get it” to have that “aha!” moment.
This was just a lovely piece of art. I mean the graphics were just unbelievable. Picky question. How long did it take you to create this masterpiece? (And if it has not been obvious, you've gained another subscriber👍)
16:04 There seems to be a lot wrong with this slide. There's no constant term in the integration. The differentiation also has the differential of y multiplied by f(x) giving the f'(x), instead of differentiation being an operator applied to f(x). Correction: The constant term is explained later in the video, so that is an understandable omission.
This unironically showed up in my recommended on the perfect moment, cheers from the army, keep up with the good work buddy, really motivated me to learn math just for the love of it!
Bravo! 👏 This is how math becomes easy! Been studying math my entire life and did engineering math throughout college. Also took a graduate level controls class for my master's where we used Fourier transforms, but THIS video right here, has done something none of those classes did. Thanks a lot for posting this. This is golden!
Thanks for real though I had some misunderstanding in calculus and trigonometry, and you clearly explained them while not making a big deal out of things that can be explained simply. Thank you again and hope you do well. Good luck with your channel and your future works. Peace!
You are amazing! Edit: Also, mathematicians are not asking "why is that useful?", because that's for engineers and physicists or computer scientists to figure out. For mathematicians it is entirely enough to say "because we can".
16:04 well not quite. Because there is no way to get back constants that were lost in the derivative. So we add a constant labeled C to represent them. WARNING‼️:NEVER forget to add constant C!!
Not exactly... In math we cant but If It is a real scenario we can, for exemple imagine a car standing still starts moving we know It acelerating at 4m/s^2 so the intregal in relation to time would be 4t + c = v but the c is the initial velocity wich is 0 so we we know v = 4t (in m/s) so we figured c.
@@everyting9240 well, look at that, you DID add a "c" there. Yes, its 0, but that's the point. You did add it. And also, in all the situations of integration, THAT IS HOW "c" IS FOUND!!! By using constraints, (and pay attention here @everything9240) not just in physics, but in maths too!!!
@@thekiwiflarethey're right though. You often have to solve for the constant using known conditions, and that's a known condition for that case so it's easy to just plug in.
@@FunctionallyLiteratePerson yeah but that completely throws out the point of the original comment - you can't know the initial conditions if all you have is the final result
Loved your trig explanation. I’ve understood that sin and cos are respective x and y points on the circle realized over time. I never realized tan is literally just slope equation. Absolutely loved that. THANK YOU SO MUCH for finally explaining the equations for sin and cos. Despite knowing the complex plain and e^i(theta)=…. I never connected it. Never tried to derive it. W video, excited to watch more
Mate, I really can’t understand how underrated your channel is, I’m mean: great editing, great voice over, a person who clearly know what he’s talking about and most importantly someone that either loves maths and science, (seeing how you’ve uploaded over 600 videos of them), or your really determined to make people understand it to a greater level. I’m a math guy myself but wow, your on another level, I planning to watch more of your videos considering how much effort has been put into them, I can’t even understand how you have such a good upload schedule. My congratulations, you have got a new subscriber and new eyes watching your amazing videos.
instant subscribe. Thank you for your very visual kind of approach to math. I've always struggled with learning math with static text book, but understand much more faster with visualization and helps with my imagination of math. Please do more videos like this so us people can learn!
The problem with math is that almost everything builds on top of another, and everything that is proven to be correct only adds to the prior knowledge, nothing correct ever becomes outdated again. Which means you cannot grasp an advanced concept without grasping many more basic concepts first and everything is only expanding more and more. There are a few fields which differ, say basic vector algebra or graph theory where the concepts are not that much related to other fundamental concepts and thus can be learned without prior knowledge of many other things, but this soon changes on advanced levels, when other branches of math are intruding these fields too. Because, the other thing about math is that everything is connected. Having a high degree of variety and a high degree of connection could be a definition of complexity. Thus advanced math inevitably gets complex.
Exactly. If you get lost on one step you're lost for all the following steps. Additionally, notation can be tricky to understand. For example he didn't explain what h means when talking about limits, so every conclusion based on anything using limits doesn't make sense to me. I don't know what f(x) means, I don't know what dx means, and I didn't understand the explanation of the integral sign. Despite "learning" how to differentiate and integrate in school I've never really understood a lot of the notation, which means I've never been able to properly understand or learn anything that builds upon things like these.
@@porkeyminch8044 It is a pity that you 'learned' differentiation/integration in school, but don't know the notation. It is hard to imagine how this can be, in fact. But I know these things from myself. Teachers are often not even aware of these things themselves. The integral sign, for instance, is just a sign for 'sum'. It is essentially an old style German 'S' letter. The sum is over a product of the value of a function labeled by the letter f at the variable position x, f(x) in notation, and the infinitesimal (infinitely small) quantity dx, x again denotes the variable, d the differential quantity. It is proven that people, who think of integration as a special kind of summation over some product terms, instead of thinking of calculating an area, for instance, have a much better grasp of the concept outside its usual context of geometry and functions of one variable. The need for a special symbol for summation is just because the summation symbol stands for something discrete, while the integral symbol stands for the same, but continuous. To explain these things, also the history of the notation, takes a few minutes, but can make a huge difference in getting familiar with it.
@@porkeyminch8044The integral sign is just a compact way of saying “this is a sum of those little rectangles which we make smaller and smaller, then add all of them to get the area under the curve” Same way 3x50 is a compact way of saying “we add the number three 50 times”
Great video, very satisfying ending, still hate the fact that you wrote sqrt(-1) which is technically undefined and -1^2 = 1^2 forgetting the parenthesis. Love from Brazil 🇫🇷
4:47 Just a minor suggestion. Perhaps avoid the combination of untextured red-green colors in your presentation so they are more color blind friendly. Suggestions: 1. Substituting one with blue or any other color combinations that are color blind friendly 2. Using differentiating textured graphics if you want to keep the red and green. (like the textured bar, columns charts in excel) Hope that helps.
I never ever intuitively understood the notation for integrals. You've opened my eyes with the "it's just telling you to sum up all rectangles with height f(x) and width dx".
That is the best Video about maths all across the Internet. BY FAR. You can feel the way you are passionate about maths and it is really enlightening. I am studying computersicence since last month and this video really motivated me to keep going.
im going to be super honest with you, this video really opened my eyes to these kinds of mathematical concepts, especially imaginary numbers! seeing things represented like this in such a fun and literal way is exactly the way i think it should be shown! ive maybe just, binge-watched 3 or 4 of your videos and ive got to say, you have huge amounts of potential, and seeing great content like this so underlooked? kinda breaks my heart! keep doing what youre doing!!!!
Exactly and people wonder why people get confused by maths, while people trying to explain it in depth make basic mistakes only to make everything exponentially more difficult within minutes while seemingly trying to beat WPM (words per minte) contest. Leaving people behind from the point basic mistakes were made.
If I had a nickel for every time MAKiT made a video about the progression of maths I would have four nickels Which is certainly a lot more than the two that Dr Doof had
Actually, roots are not the inverse operation of exponents, logarithms are the inverse operation, roots is just a type of exponent represented in another way. And you might argue that division is another way of expresing a kind of multiplication, but it is actually a collection of substractions, so its actually the inverse, but thats just not the case for roots. With functions you see it clearly, f(x)=2^x the inverse function is g(x)=log2(x)
Same same at 23:20 Csc(x)/Sec(x)/Cot(x) are not the inverse of Sin(x)/Cos(x)/Tan(x), those are just the multiplicative inverse. The arc functions are the inverse functions.
Really good video! They tell us the Notation in school und university, but in my case for example they don’t dive so deep in the question “why is it like that?” in and u just answered this question with this video! Thanks
I suggest you watch brain nourishment There's a guy making brainrot videos that talk about math, I don't remember the name, but he's really funny. You can look up one of his videos though (Jenna Ortega teaches u substitution or Taylor Swift explains the Taylor series)
This may be the most densely packed math video I've seen and I'm an avid watcher of 3b1b but somehow it's also the easiest to understand. Great work dude!
15:35 The sigma Σ in repeated addition btw does just stand for S as in "sum" (a different word for addition) The Integral symbol ∫ is also just an S. This time from an old way of writing the letter s in cursive (known as the "long s") and again just stands for "sum"
already attending university for mathematics, but this video really makes me fall in love again with the subject. Thank you for sharing the beauties of mathematics with the world ❤
This is the most comprehensive and easy to understand way to describe how maths builds upon itself that I have ever seen. You even introduced cyclical functions correctly. I almost wonder if it would be beneficial to introduce modular arithmetic before doing cyclical functions, but it might come out of left field for some students. Plus it would make the video longer. Teaching maths is a big challenge
While I haven’t learned anything new from this video, a few years ago my mind would have been blown. I almost learned something like I never knew the name of the variable In the Fourier transform, and I don’t know how to google that symbol. But then you just ignored it. I use math a lot, I commonly use trig in programming and even occasionally calculus with derivatives and antidirivitives. Yet I still watched the video all the way through so you were still entertaining enough, even without me learning anything (though I did just watch most of it on 2x speed) can you tell me the name of that variable, my textbook doesn’t tell me it just shows the symbol.
@@bjornfeuerbacher5514 Actually 🤓🤓 that is a bit unnecessarily pedantic. But if you want to be pedantic then you shouldn't write "xi" either - that is not an official transcription. Anyway it's all explained in detail in the wikipedia link.
@@Simchen What do you mean? As I understand the Wikipedia article, "xi" is the official transcription...? And I wouldn't call it pedantic to distinguish between uppercase and lowercase letters.
I’ve always wanted videos that show the relationships between multiple concepts all in one format, but I couldn’t find any, so I created a channel in my native language to do just that. The amount of work it takes-even without editing-just to choose what to cover, is insane. Now, there’s this awesome channel that does it with 3D modeling, exactly how I imagined it in the ideal version. The internet is amazing-so glad this channel exists!
I know this is overkill but the context in the derivative part is perfect to go a little more formal so the problem with taking the slope of 2 points is that the function might do weird stuff in between them, but notice how after the green dot go past all the weird stuff it become a much better value for the slope so now we can choose to just use the slope value between two points as long as there is no weird movement in between them but how do we define this "weird movement"? now thats hard, we cant just say "there shall be no movement between the two points we are checking" because that would either make this tool only useful when slope=0 or those are the same point, which cause 0/0 again now think about what this "weird movement" actually is, its a group of values that have a much larger differences to the value of the two points we are checking compare to the points that are closer to the two points, so what if we just specify a maximum difference that we feel like should be close enough? 0.01, 0.0001, and if all of the values from point 1 to point 2 fall within this range we would take the slope value, if not we move them closer until they does, and if it somehow just can never fall within the range we just say the slope doesnt exist, but there is a problem, what if the two values are 0.0000001 and 0.0000002? then a difference of 0.0005 might not seem big when we specify the range 0.01 but is now way out of proportion compare to the two point we are checking and now, to resolve that issue, we need one fact about the real number, there are always numbers in between any two numbers (notice that this imply there are infinitely many numbers in between any two number as given 2 numbers a------b, you can get 1 number in between them a----c----b, and because c is also a number you can also get more number from it and a and b a--d--c--e--b, .....), and because there are always numbers in between any two numbers, you can keep getting closer to a number (reducing a larger number) without ever being exactly equal or lesser than the number you are getting to and with that fact we can now fix the proportion issue, notice how if there is a range where all the value between the two point would all fall into that range, those values would also satisfy any larger difference (difference of 0.1 and 0.2 is 0.1, the difference is within the range of size 1, which mean the difference is also within range of size 2.34, of size 5.69, ...) so we just need to specify a range of differences that are all larger than a certain value rather than focusing on a single one (this make defining it easier), if we set any specific value there will eventually be 2 points on a function that are close to each others but still have weird jump in between them, but we know we can keep getting smaller and smaller while never reaching the number, so now we just need to verify one final thing, does the "weird movement" move with us as the two point get close? i would say no, and if it does for some reason, we will just say the slope doesnt exist in that case, as the two points are getting closer, they will eventually go past the "weird movement" as they are static while we are moving, so we can say that there should be no differences of the value within any range that is larger than 0, or "for all range difference > 0", i will now start calling them "e" for short, so "for all e > 0", but now we get a new prolem, since we say the differences have to fall within the range e "for all e > 0" and between any two specific and distinct points there would be a difference larger than 0, that would make this tool useless since its false for pretty much all cases, so we should loosen the condition a little, the issue here is that for any specific two point we can keep decreasing e so that it will be smaller than the difference between those two points (which is also the property that make us use it in the first place), going back to when we still talk about a specific e value like 0.01, 0.0001, ... the reason it felt intuitive and make sense is because we can feel like it being useful, for any sane function that map actual real life useful numbers, it wont be arbitrarily large or small but within a reasonable range so if we specify a decently small range we could capture the essence of "getting the two points close enough so the there isnt any weird movement between them", the only reason it doesn't work is because of the proportion but wait, the reason that didnt work is because the range e then was static so any two values could eventually be close enough that the "weird movement" fall within the range e but is still proportionally huge compare to the two values, but now our range e is not static anymore, in fact our current problem is that it keep getting too small to be useful, so what if we find a distance between the two points in relation to e? now our distance is also getting smaller as e get smaller, if there are any "weird movement" that goes outside of e, we reduce our distance, which i will call "d" from now on for short, we reduce d until the movement is within e again and then keep getting smaller, if for some reason, no matter how much we reduce d we just cant get it to fall within e, we just say that the slope doesnt exist in that case now we have "for each e > 0, if there is a value d that is the distance between two points, and all the values in between the two points fall within e (or |x1 - x2| = d, for all a in between x1 and x2 => |f(a) - f(x1)| < e and |f(a) - f(x2)| < e), if the condition is satisfied, then we accept the slope between the two values as the value that represent "how much the function is changing" but in a single function, how much its changing change through out the entire function, so we want to be able to specify which part of the function we want to get the "changing factor", ideally we want to be able to do this for all real number offset on the function, so lets do just that, instead of two arbitrary point x1 and x2, now we specify the position we want to check with a, and x1, x2 will be around where a is, we can even align them so that a is perfectly in the center of x1 and x2, which would make them look like this a-(d/2), a+(d/2), they now have distance d and surround a, but recall how we use the trick "for all e > 0" because if something satisfy the current e range, then it also satisfy any larger e range so we doesnt have to say something like "with an e that is infinitely close to 0", turned out we can do the same thing again, since for any range d surrounding a that make all the points inside them have a value within the current range e, the same thing would hold if d is any smaller, so we can say "exist d > 0 such that ...", and because d is also decreasing, if at a certain d its true, it will also hold for d/2, so a-(d/2), a+(d/2) could be simpified to a-d, a+d (just scale down d until its equal to the previous d/2), and in range notation (a-d, a+d), and since we are now centering around a, we doesnt need to check for e range around both of the border anymore and instead just check the range on a, so for all x within range (a-d, a+d) the condition it must hold is that |g(x) - L| < e with L being the value of the "rate of change" and g(x) is the rate of change with respect to a (using normal slope), so if a number L exist such that all the point surrounding it with distance d has the rate of change difference to L fall within e, that is the rate of change of the function at point a or "for any ϵ>0 , we can find a δ > 0 such that if 0
In your complex number section, you really shouldn't have said that -1^2 = 1^2; that's too fundamental mistake to let it slip by. It must read (-1)^2 = 1^2.
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This is a sign to finish my math homework.
Procrastinate to watch MAKiT video
@@MozzarellaWizard FR
Fr same here
It's a _sine_ to finish your math homework.
Same
“If no real number would work, than how about we just IMAGINE one?” Beautiful transition
Then
@@AlexandroGarcia6492 aren't you pretentious, I bet you get mad when people end a sentence with a preposition.
Thank you. I was explaining to my friend why in "order of operations" that multiplication and division were interchangeable same with addition and subtraction. And when i said "because basically theyre the same thing" she looked at me as if i was crazy 😂
For me things like that help a lot with understanding more complex topics.
They’re the same thing in the sense they’re opposites or inverses of each other, important detail
Bottom line is they commute
The order of operations is an arbitrary convention
@@savazeroa try solving 5+2*6 solely going left to right without using order of operations.
That's how many eggs I collected from the chooks over the past 3 days, so there is a correct answer:
17
Bro just went from addition to a Fourier transform in 40 minutes and made it understandable
One of the best math videos I've ever seen
Couldn’t agree more. If anyone is seeing this and wants to skip because he starts off by describing arithmetic, I urge you to stick around.
This was a comical account of information packed into one single video... and I'm here for it!
After so, so many streams. It is finally here. The 40min math vid.
when i saw the thumbnail i was like "pshhhhh, math isn't difficult" but then when i pressed the video and saw the first equation i said "nvm"
Not really, it's just notation here. Fourier Transform isn't so hard to understand.
@@virtueose The Laplace Transform? Yeah.
@@SbF6H the thing with these equations is that, if you dont know what it represents, its very difficult to reverse engineer what it represents even if you know the notation unlike some simpler equations. i personally didnt know it but its pretty easy to understand.
fourier, laplace generalizes to all complex values of frequency, fourier only generalizes to those with 0 real component @@SbF6H
@@badabing3391 What do you mean? I was just shoving in real values into FFT and getting my work done perfectly.
A future original 3B1B in the making, keep up the great work and amazing videos. Would love to see longer videos if it meant minimizing holes and gaps. Thank you for your work!
he's 3b1b but with rgb lol
@@warriorofhyperborrea gamer 3b1b
3B1B knows mathematics quite well. This guy doesn't know even basic things.
@@diogeneslaertius3365 what did he sayyyyyyyyyyyyy
@@Shadoxite from my other commentary (only watched a few minutes and stopped): "Tetration is useful in combinatorics and pure mathematics is not about applications.
Applications are easy and they will always show up sooner or later. Root and 1/p power are not the same operations. Only if you match the domains specifically and pick a certain root, you could somewhat bring them together say for R+. They are two very different beasts if you properly consider complex numbers. Why are you confusing people if you don't understand the basics yourself?"
I am so glad that Im getting an explanation for complex math, my teachers all basically taught me was "this is how you use them because this is how they are used" but now why they are used that way or what they are, and for me to learn I have to actually understand something at a fundamental level
Got to hand it to you mate, although i knew these concepts beforehand, the visualization and most importantly your explanations were amazing, very underrated video, amazingly put
"\left(
ight)" so your parenthesis stretch to the height of the thing inside
Also \sin \cos and so on to make those operators not cursive
\left(\! \!
ight)
if the space between the interior expression and each parenthesis is too wide
\qty from physics library is good alternative for \left
ight
also usually \mathrm{d} is used for derivatives
@@okicek3016 That's not cursive, it's italics.
I always find it so odd that people struggle so much with algebra. Probably a result of it being taught way too late, as substitution is so basic that it really should just be taught around the same time as multiplication (and should be followed within a year or two by parentheses and factorisation, as they're another one people tend to struggle with due to how late they're taught.)
Probably failed by american education lol /s
I see people struggling with Fractions, it's so easy, it's literally just division and people struggle with it, in my opinion they should only teach fractions and avoid pure division as much as possible, because in the future(High school) these people won't use "÷" anymore and will only use fraction.
@@reclaimer2019 you could probably teach ÷ when teaching other alternative notations like *, ^, and ↑↑↑ and just teach them like you would alternative characters in English like @, &, etc.
Though you can always just teach both division and fraction notation simultaneously as different was of writing it, as ÷ is really important for factorisation, as 1/x(2+3) [2*(1/x)+3*(1/x)] and 1÷x(2+3) [1÷(2x+3x)] aren't the same thing [x=1, 1/1(2+3)=5, 1÷(2*1+3*1)=1/5]. You Can get around this with 1/(x(2+3)) or a long fraction sign that I don't feel like looking for the unicode for, but a division sign does the job just fine too.
@@reclaimer2019they should be taught that these are equal, also, the notation for a single line equation can get very messy, but it makes absurd sense.
Like how 1/1+1 is different than 1/(1+1), but some people seem to not be able to recognize this.
People struggle with algebra due to the fact it makes no sense. This is because algebra in Western countries isn't taught systematically but with an adhoc approach. When we were going over equations we never went over what operations you can do to them.
Also parentheses aren't explained well usually. For example something like this 5+(5-4) would be "incorrect" to solve as 5+5-4=6 even though the parenthesis in this case do nothing.
"Moles are not a unit!!"
Dude I felt that to my core I absolutely despised conversions 😂😂😂
I love these videos! Also 25:01 I recommend you enclose the -1 in parentheses or else it is -(1)^2 = -1
i was looking for this comment😋
I love why math works and I’m glad more people are covering it in depth. You should do mechanics next, it’s pretty easy to explain how we get the laws of motion and why things like energy are useful
yea that should be fun to watch
Mechanics would be sweeeet!!!
YESS!! Mechanics would be a fun video!
Im finishing my thesis for my undergrad in physics. Never seen such a beautiful explanation for the Fourier transform.
Amazing vid. :)
Not exactly sure why I watched the entire video, considering I've done all that in depth throughout my academic journey, but damn, that's an easy to grasp and extremely quick explanation to lots of interchanging mathematical concepts that I was taught through years of math classes. Honestly well done. Had this existed half a decade ago, it would have made my life way more "understandable" (definitely not easier - applying everything mentioned here to actual use is why proper education takes years, not 30 minutes).
Would have liked a bit of a deeper dive into Polar coordinates though, considering how useful they are throughout disciplines.
Quick note at 16:00,
dy/dx is actually the derivative f’(x)
Whereas if we want to do the action of taking the derivative of f(x),
We gotta write out d/dx f(x).
Think of d/dx as the derivative operator,
Just like how x tells us to multiply,
d/dx tells us to take the derivative
While dy/dx = f’(x)
d(f(x)) /dx=f'(x) =dy/dx
30:43 thank you man. i feel so validated. i tried explaining to everyone i could that sines and cosines just don't feel usable. un-graspable and undefined. but here they are. in their true form. beautiful.
another way to write sin and cos:
sin(z) = (e^(iz) - e^(-iz))/(2i)
cos(z) = (e^(iz) + e^(-iz))/2
This format makes them easier to use with complex inputs z, can help you prove derivative and integral trig properties, as well as shows the connection to the hyperbolic trig functions sinh and cosh.
This didn't sit right with me and i kept mentioning it during the stream this was being made
I personally would define sin and cos by their infinite taylor series,
of course, the formula for the taylor series requires the derivatives of sin and cos respectively, but in the case of sin and cos they're nice infinite sums (for the maclauren series)
technically, i think maybe this is a circular definition as the motivation behind taylor series involves the derivatives of sin and cos, and we're using that to define sin and cos, but i can't think of anything better-
Defining them in terms of complex exponentiation would require a definition of complex exponentiation
If you define complex exponentiation by plugging i into the taylor series of e^x, and then proving e^ix is equal to cos(x)+isin(x),
(using the taylor series of cos(x) and sin(x))
you're still using the taylor series.
if you don't want to use the taylor series, and just define complex exponentiation by euler's formula, you still have cos and sin in eulers formula! it's a circular definition!
Please tell me where i'm wrong- i think i'm probably wrong
@@savazeroa @savazeroa no you're 100% correct, i noticed that in the vid as well that it seemed self-referntial and kinda reduntant but i guess he didn't wanna go on a tangent to explain series but yh defining them with their series expansion would be more correct than what is shown
derivatives came way earlier than I thought they would
There was a lot of maths to fit into 40 mins (and yet it still took me 4 mins to explain division)
The derivate of sin and cosin osciolating but never changing property now makes so much sense considering theyre made up of exponetionals. Thanks for revealing some deep math knowledge that will help me in my Electrical escapades.
you lost me at 0:28
I learned that at 3 years old bro
Ewolkmind
I'm a year 10 student in the UK and this is actually a simplified explanation with good animation. Good work.
I have been struggling with my digital signals and systems course because I was afraid of notation, and I did not completely understand the transition from complex numbers and euler's formula into the Fourier Transform. It's the day before my second midterm, and this video might help me save my grade. Thank you so much, and please make more videos like this to help us engineering students!
Thought I was just watching this for funsies. *Sweets in 1st yr engineering*
The phrase "This is simple" made me entirely give up on math and the sciences as a whole.
This gives it a little more hope lol.
As a maths enjoyer, I have no Idea what a normal person would think watching this...
But for me, I absolutely love this content! You display it very well.
This is AMAZING. Thank you for making it. I've just finished an AP math course (basic 1st year math in hs ) and this went through and beyond all my knowledge 😅
The problem is that math is explained to fast. Teachers move on to the next subject before the previous one is understood.
In some cases, i feel like math is too easy, like log, I would understand it in 6th grade, alongside with exponents, and log is taught in 10th grade.
@@Snakehandler268 something funny i understood everything in this video and only watched it twice im in 8th grade and my teachers face when i started explaining Fourier transform
Perhaps. But my opinion is that people just aren’t required to, or simply don’t, do enough problems on their own. Math isn’t something you can cram for just before an exam, like, say, history. You need to develop an intrinsic understanding that practically always takes a lot of time, effort and practice.
@chudleyflusher7132 This is true and if you stop using it after high school you will forget things quickly. I had to relearn algebra from the ground up 2 years after I graduated high school cause I took a gap year and solved zero math.
That was honestly the most well explained video on maths I have ever seen. My friends often struggle to understand why I find maths so exciting, but I'm pretty sure they'll understand once they watch this. I loved the flow of introducing all of the topics, as well as the animations which made it super easy to not get lost in all of the new words for someone whose first language isn't English. Thank you so much for this wonderful piece of media contributed to the internet, I'll make sure to recommend your channel to as many people as I can.
This video was amazing!! It’s like you distilled all of math and TH-cam to a 40 minute thesis. It was well worth the effort in my opinion.
A lot of math is missing from this, continue to explore!
Oh my friend you have much to explore, it will be the most fucked up, most beautiful endeavor you could ever peer into. Have fun!
This is math from the 19th century at most, so very much not all of math. The things people are currently doing in math is a lot lot lot more complicated.
It was so jarring to be introduced to math like "Divide rise by run DO IT DO IT NOW OR YOU GET BAD GRADE" instead of explaining it more organically like this.
This is the simplified visual explanation I needed during math classes. THANK you!
Hi Matik! I've just begun trigonometric, it's hard to approach at the begin, but your video makes me more certainty about mathematic.Great video and greetings from Italy!!!
The intuition for the Fourier transform was really satisfying.
Your comment made me actually watch that bit of the video and wow! What an amazing intuition.
Top 5 ytbers imo, and remember, aside from Vsauce, this is the only guy that does anything academic in the top 5
I think you have no idea how good ur vids are. Now if you did this with physics THEN I actually straight up explode
I am planning on MAKiNG "How Physics Becomes Difficult" and "How Chemistry Becomes Difficult"
@@MAKiTHappen YES YES YES YES YES YES YES
I've messed with all of these functions and haven't felt like I've ever had a better understanding then right now after watching this video. I'm sure the average person will need more so please keep up the incredible work that you're doing!
Linear algebra explained in a step-by-step fashion would be amazing.
for a lack of better word, this channel is criminally underrated.
Excellent video. For the sake of simplicity, we're often taught these topics without any further explanation as to how they were derived and where they were derived from which can leave one with a lot of questions. This video does a great job laying it all out with fun graphics. Subbed!
Bro really taught math to an alien
This is very useful because it's actually applicable math I can use in the real world
It seems like bro wanted to flex his maths skills in front of us and he did a really good job
Bro summarised my bachelor's maths in 40 minutes, too bad it's 2 years too late.
24:57 but the term -1² doesnt equal to 1 it equals to -1 because first you square -1 and then multiply it by negative, you should have writed (-1)² which equals to 1
This was… incredible!! I absolutely love your videos and how you build up concepts. Your visuals are spectacular and your explanations show an amazing and unique ability to communicate concepts in a way that is absolutely perfect for anyone who just feels like “they don’t get it” to have that “aha!” moment.
This was just a lovely piece of art. I mean the graphics were just unbelievable.
Picky question. How long did it take you to create this masterpiece? (And if it has not been obvious, you've gained another subscriber👍)
3 weeks in total. Around 200-300 hours of work
@@MAKiTHappenthats crazy thanks for the video. Your ability to simplify complex topics is amazing
@@MAKiTHappenrespect
@MAKiTHappen it's Blender? When do you sleep?
I went from barely understanding 6th grade math to harvard math 55 level from this video.. thankyou!
16:04 There seems to be a lot wrong with this slide. There's no constant term in the integration. The differentiation also has the differential of y multiplied by f(x) giving the f'(x), instead of differentiation being an operator applied to f(x).
Correction: The constant term is explained later in the video, so that is an understandable omission.
This unironically showed up in my recommended on the perfect moment, cheers from the army, keep up with the good work buddy, really motivated me to learn math just for the love of it!
The narrator and the editor deserves an award
Thank you
Bravo! 👏 This is how math becomes easy!
Been studying math my entire life and did engineering math throughout college. Also took a graduate level controls class for my master's where we used Fourier transforms, but THIS video right here, has done something none of those classes did.
Thanks a lot for posting this. This is golden!
This is the best math video I’ve ever seen
Just started Uni as a STEM Student and with minimal math knowledge, this helped a lot,
Thank you!
bro thinks we wouldn't notice the rad joke
Thanks for real though I had some misunderstanding in calculus and trigonometry, and you clearly explained them while not making a big deal out of things that can be explained simply. Thank you again and hope you do well. Good luck with your channel and your future works. Peace!
You are amazing!
Edit: Also, mathematicians are not asking "why is that useful?", because that's for engineers and physicists or computer scientists to figure out. For mathematicians it is entirely enough to say "because we can".
I got all the way to (-1)^.5 =i in high-school and wish I kept learning. The foier transform looks neat
16:04 well not quite. Because there is no way to get back constants that were lost in the derivative. So we add a constant labeled C to represent them.
WARNING‼️:NEVER forget to add constant C!!
Not exactly... In math we cant but If It is a real scenario we can, for exemple imagine a car standing still starts moving we know It acelerating at 4m/s^2 so the intregal in relation to time would be 4t + c = v but the c is the initial velocity wich is 0 so we we know v = 4t (in m/s) so we figured c.
@@everyting9240 that's physics
@@everyting9240 well, look at that, you DID add a "c" there. Yes, its 0, but that's the point. You did add it. And also, in all the situations of integration, THAT IS HOW "c" IS FOUND!!! By using constraints, (and pay attention here @everything9240) not just in physics, but in maths too!!!
@@thekiwiflarethey're right though. You often have to solve for the constant using known conditions, and that's a known condition for that case so it's easy to just plug in.
@@FunctionallyLiteratePerson yeah but that completely throws out the point of the original comment - you can't know the initial conditions if all you have is the final result
Loved your trig explanation. I’ve understood that sin and cos are respective x and y points on the circle realized over time. I never realized tan is literally just slope equation. Absolutely loved that.
THANK YOU SO MUCH for finally explaining the equations for sin and cos. Despite knowing the complex plain and e^i(theta)=…. I never connected it. Never tried to derive it.
W video, excited to watch more
15:58 The notation is not quite right. dy/dx is a derivative, but derivative of f(x) is d/dx f(x).
yeah dy/dx is implicit differentiation 🤦
differentiate y with respects to x treating y as a function of x
Came to the comments to comment this. Thanks for the good work.
No it was a typo.
He wrote dy/dx f(x) which means we're differentiating y with respect to x and then multiplying it with f(x).
@@powercables
@@rnd_penguin yes, if there is a y, it is multiplication but not differentiating f(x).
Mate, I really can’t understand how underrated your channel is, I’m mean: great editing, great voice over, a person who clearly know what he’s talking about and most importantly someone that either loves maths and science, (seeing how you’ve uploaded over 600 videos of them), or your really determined to make people understand it to a greater level.
I’m a math guy myself but wow, your on another level, I planning to watch more of your videos considering how much effort has been put into them, I can’t even understand how you have such a good upload schedule.
My congratulations, you have got a new subscriber and new eyes watching your amazing videos.
24:59 supposed to be (-1)^2 = 1. Great video !
yea
i swear my mind was panicking when i didn't see the parenthesis.
Division has one critical non-operation : division by zero; addition, multiplication and subtraction can handle zero in all case.
1:45 freaked me out wth
Me too bruh wtf
instant subscribe. Thank you for your very visual kind of approach to math. I've always struggled with learning math with static text book, but understand much more faster with visualization and helps with my imagination of math. Please do more videos like this so us people can learn!
The problem with math is that almost everything builds on top of another, and everything that is proven to be correct only adds to the prior knowledge, nothing correct ever becomes outdated again. Which means you cannot grasp an advanced concept without grasping many more basic concepts first and everything is only expanding more and more. There are a few fields which differ, say basic vector algebra or graph theory where the concepts are not that much related to other fundamental concepts and thus can be learned without prior knowledge of many other things, but this soon changes on advanced levels, when other branches of math are intruding these fields too. Because, the other thing about math is that everything is connected. Having a high degree of variety and a high degree of connection could be a definition of complexity. Thus advanced math inevitably gets complex.
Exactly. If you get lost on one step you're lost for all the following steps.
Additionally, notation can be tricky to understand. For example he didn't explain what h means when talking about limits, so every conclusion based on anything using limits doesn't make sense to me. I don't know what f(x) means, I don't know what dx means, and I didn't understand the explanation of the integral sign. Despite "learning" how to differentiate and integrate in school I've never really understood a lot of the notation, which means I've never been able to properly understand or learn anything that builds upon things like these.
@@porkeyminch8044 It is a pity that you 'learned' differentiation/integration in school, but don't know the notation. It is hard to imagine how this can be, in fact. But I know these things from myself. Teachers are often not even aware of these things themselves. The integral sign, for instance, is just a sign for 'sum'. It is essentially an old style German 'S' letter. The sum is over a product of the value of a function labeled by the letter f at the variable position x, f(x) in notation, and the infinitesimal (infinitely small) quantity dx, x again denotes the variable, d the differential quantity. It is proven that people, who think of integration as a special kind of summation over some product terms, instead of thinking of calculating an area, for instance, have a much better grasp of the concept outside its usual context of geometry and functions of one variable. The need for a special symbol for summation is just because the summation symbol stands for something discrete, while the integral symbol stands for the same, but continuous. To explain these things, also the history of the notation, takes a few minutes, but can make a huge difference in getting familiar with it.
@@porkeyminch8044The integral sign is just a compact way of saying “this is a sum of those little rectangles which we make smaller and smaller, then add all of them to get the area under the curve”
Same way 3x50 is a compact way of saying “we add the number three 50 times”
Your amazing! This has been a great watch start till end. We all appreciate the time taken to make the animations too
Great video, very satisfying ending, still hate the fact that you wrote sqrt(-1) which is technically undefined and -1^2 = 1^2 forgetting the parenthesis.
Love from Brazil 🇫🇷
Your videos are extremely well produced, I love the look of them. I will be using them for my students.
God Bless. Keep up the great work MAKiT
20:27LMAOOOO "we'll stick to radians because they are just so RAD"😂😂😂😂😂😂😂 im dying of laughters
wasn't that funny tbh
Enviable comedy bar
what a FANTASIC video! Clearly explained and gorgeous visuals, making something that seemed impossible, possible. Thank you
4:47 Just a minor suggestion. Perhaps avoid the combination of untextured red-green colors in your presentation so they are more color blind friendly. Suggestions:
1. Substituting one with blue or any other color combinations that are color blind friendly
2. Using differentiating textured graphics if you want to keep the red and green. (like the textured bar, columns charts in excel)
Hope that helps.
Why?
@@xinpingdonohoe3978Because some people are colour blind.
Most colourblind people cant distinguish red and green@@xinpingdonohoe3978
Very well made. Can't even imagine the amount of work that must have went into this.
I never ever intuitively understood the notation for integrals. You've opened my eyes with the "it's just telling you to sum up all rectangles with height f(x) and width dx".
No teacher ever told you that this is the meaning behind the notation?!? :O You had rather bad teachers. :(
@@bjornfeuerbacher5514 Maybe they did, but it was never made clear to me x)
That is the best Video about maths all across the Internet. BY FAR. You can feel the way you are passionate about maths and it is really enlightening. I am studying computersicence since last month and this video really motivated me to keep going.
the two guys chatting in the live chat is more entertaining than the video itself
im going to be super honest with you, this video really opened my eyes to these kinds of mathematical concepts, especially imaginary numbers! seeing things represented like this in such a fun and literal way is exactly the way i think it should be shown! ive maybe just, binge-watched 3 or 4 of your videos and ive got to say, you have huge amounts of potential, and seeing great content like this so underlooked? kinda breaks my heart! keep doing what youre doing!!!!
0:57 yes you can. You can imagine the number on the y axis repeated as many times as however big the number on the x axis is and vice versa
Exactly and people wonder why people get confused by maths, while people trying to explain it in depth make basic mistakes only to make everything exponentially more difficult within minutes while seemingly trying to beat WPM (words per minte) contest. Leaving people behind from the point basic mistakes were made.
Yeah but how do you repeat a number 2.2 times though?
@@IndianGeek5589 you repeat it 2.2 times
@@tombullish3198 wait you understood what I meant? Great, I thought I was kinda babbling but I’m happy u understand bro😁
@ yeah but how do you represent the number 4, 2.2 times, without using multiplication.
Incredible Video, good job!
If I had a nickel for every time MAKiT made a video about the progression of maths I would have four nickels
Which is certainly a lot more than the two that Dr Doof had
Actually, roots are not the inverse operation of exponents, logarithms are the inverse operation, roots is just a type of exponent represented in another way.
And you might argue that division is another way of expresing a kind of multiplication, but it is actually a collection of substractions, so its actually the inverse, but thats just not the case for roots.
With functions you see it clearly, f(x)=2^x the inverse function is g(x)=log2(x)
Same same at 23:20
Csc(x)/Sec(x)/Cot(x) are not the inverse of Sin(x)/Cos(x)/Tan(x), those are just the multiplicative inverse.
The arc functions are the inverse functions.
16:05 wrong operation, true: d(f(x))/dx = f'(x) , no (dy/dx)*f(x) = f'(x) , what is y in this case??
Really good video! They tell us the Notation in school und university, but in my case for example they don’t dive so deep in the question “why is it like that?” in and u just answered this question with this video! Thanks
this can fix my brain rot
I suggest you watch brain nourishment
There's a guy making brainrot videos that talk about math, I don't remember the name, but he's really funny. You can look up one of his videos though (Jenna Ortega teaches u substitution or Taylor Swift explains the Taylor series)
This is so amazing! Seeing how things connect made a lightbulb go off in my head
2:36 Shouldn't this be ^a and not ^b?
This may be the most densely packed math video I've seen and I'm an avid watcher of 3b1b but somehow it's also the easiest to understand. Great work dude!
15:35
The sigma Σ in repeated addition btw does just stand for S as in "sum" (a different word for addition)
The Integral symbol ∫ is also just an S. This time from an old way of writing the letter s in cursive (known as the "long s") and again just stands for "sum"
heh sigma
already attending university for mathematics, but this video really makes me fall in love again with the subject. Thank you for sharing the beauties of mathematics with the world ❤
29:44 its okay makit, we dont wnat to put more on your plate
I really appreciate this as it puts it in terms that connects and makes it easier to comprehend. A-levels look easier with this videl man.
Watching this video and not completely wrapping my head around is like looking at a post game area in a game which you can't reach yet
This is the most comprehensive and easy to understand way to describe how maths builds upon itself that I have ever seen. You even introduced cyclical functions correctly. I almost wonder if it would be beneficial to introduce modular arithmetic before doing cyclical functions, but it might come out of left field for some students. Plus it would make the video longer. Teaching maths is a big challenge
This is why I love math
Awesome work on this one! Exactly what I needed to push harder in my studies. Keep up the good work! Your videos are legendary.
While I haven’t learned anything new from this video, a few years ago my mind would have been blown. I almost learned something like I never knew the name of the variable In the Fourier transform, and I don’t know how to google that symbol. But then you just ignored it. I use math a lot, I commonly use trig in programming and even occasionally calculus with derivatives and antidirivitives. Yet I still watched the video all the way through so you were still entertaining enough, even without me learning anything (though I did just watch most of it on 2x speed) can you tell me the name of that variable, my textbook doesn’t tell me it just shows the symbol.
You mean ξ ? That's Xi.
en.wikipedia.org/wiki/Xi_(letter)
@@Simchen Actually, it's xi.
Xi (uppercase) is Ξ.
@@bjornfeuerbacher5514 Actually 🤓🤓
that is a bit unnecessarily pedantic. But if you want to be pedantic then you shouldn't write "xi" either - that is not an official transcription.
Anyway it's all explained in detail in the wikipedia link.
@@Simchen What do you mean? As I understand the Wikipedia article, "xi" is the official transcription...?
And I wouldn't call it pedantic to distinguish between uppercase and lowercase letters.
It can be any variable you want, but common practice is xi as pointed out here
I’ve always wanted videos that show the relationships between multiple concepts all in one format, but I couldn’t find any, so I created a channel in my native language to do just that. The amount of work it takes-even without editing-just to choose what to cover, is insane. Now, there’s this awesome channel that does it with 3D modeling, exactly how I imagined it in the ideal version. The internet is amazing-so glad this channel exists!
I know this is overkill but the context in the derivative part is perfect to go a little more formal
so the problem with taking the slope of 2 points is that the function might do weird stuff in between them, but notice how after the green dot go past all the weird stuff it become a much better value for the slope
so now we can choose to just use the slope value between two points as long as there is no weird movement in between them
but how do we define this "weird movement"? now thats hard, we cant just say "there shall be no movement between the two points we are checking" because that would either make this tool only useful when slope=0 or those are the same point, which cause 0/0 again
now think about what this "weird movement" actually is, its a group of values that have a much larger differences to the value of the two points we are checking compare to the points that are closer to the two points, so what if we just specify a maximum difference that we feel like should be close enough? 0.01, 0.0001, and if all of the values from point 1 to point 2 fall within this range we would take the slope value, if not we move them closer until they does, and if it somehow just can never fall within the range we just say the slope doesnt exist, but there is a problem, what if the two values are 0.0000001 and 0.0000002? then a difference of 0.0005 might not seem big when we specify the range 0.01 but is now way out of proportion compare to the two point we are checking
and now, to resolve that issue, we need one fact about the real number, there are always numbers in between any two numbers (notice that this imply there are infinitely many numbers in between any two number as given 2 numbers a------b, you can get 1 number in between them a----c----b, and because c is also a number you can also get more number from it and a and b a--d--c--e--b, .....), and because there are always numbers in between any two numbers, you can keep getting closer to a number (reducing a larger number) without ever being exactly equal or lesser than the number you are getting to
and with that fact we can now fix the proportion issue, notice how if there is a range where all the value between the two point would all fall into that range, those values would also satisfy any larger difference (difference of 0.1 and 0.2 is 0.1, the difference is within the range of size 1, which mean the difference is also within range of size 2.34, of size 5.69, ...) so we just need to specify a range of differences that are all larger than a certain value rather than focusing on a single one (this make defining it easier), if we set any specific value there will eventually be 2 points on a function that are close to each others but still have weird jump in between them, but we know we can keep getting smaller and smaller while never reaching the number, so now we just need to verify one final thing, does the "weird movement" move with us as the two point get close? i would say no, and if it does for some reason, we will just say the slope doesnt exist in that case, as the two points are getting closer, they will eventually go past the "weird movement" as they are static while we are moving, so we can say that there should be no differences of the value within any range that is larger than 0, or "for all range difference > 0", i will now start calling them "e" for short, so "for all e > 0",
but now we get a new prolem, since we say the differences have to fall within the range e "for all e > 0" and between any two specific and distinct points there would be a difference larger than 0, that would make this tool useless since its false for pretty much all cases, so we should loosen the condition a little, the issue here is that for any specific two point we can keep decreasing e so that it will be smaller than the difference between those two points (which is also the property that make us use it in the first place), going back to when we still talk about a specific e value like 0.01, 0.0001, ... the reason it felt intuitive and make sense is because we can feel like it being useful, for any sane function that map actual real life useful numbers, it wont be arbitrarily large or small but within a reasonable range so if we specify a decently small range we could capture the essence of "getting the two points close enough so the there isnt any weird movement between them", the only reason it doesn't work is because of the proportion
but wait, the reason that didnt work is because the range e then was static so any two values could eventually be close enough that the "weird movement" fall within the range e but is still proportionally huge compare to the two values, but now our range e is not static anymore, in fact our current problem is that it keep getting too small to be useful, so what if we find a distance between the two points in relation to e? now our distance is also getting smaller as e get smaller, if there are any "weird movement" that goes outside of e, we reduce our distance, which i will call "d" from now on for short, we reduce d until the movement is within e again and then keep getting smaller, if for some reason, no matter how much we reduce d we just cant get it to fall within e, we just say that the slope doesnt exist in that case
now we have "for each e > 0, if there is a value d that is the distance between two points, and all the values in between the two points fall within e (or |x1 - x2| = d, for all a in between x1 and x2 => |f(a) - f(x1)| < e and |f(a) - f(x2)| < e), if the condition is satisfied, then we accept the slope between the two values as the value that represent "how much the function is changing"
but in a single function, how much its changing change through out the entire function, so we want to be able to specify which part of the function we want to get the "changing factor", ideally we want to be able to do this for all real number offset on the function, so lets do just that, instead of two arbitrary point x1 and x2, now we specify the position we want to check with a, and x1, x2 will be around where a is, we can even align them so that a is perfectly in the center of x1 and x2, which would make them look like this a-(d/2), a+(d/2), they now have distance d and surround a, but recall how we use the trick "for all e > 0" because if something satisfy the current e range, then it also satisfy any larger e range so we doesnt have to say something like "with an e that is infinitely close to 0", turned out we can do the same thing again, since for any range d surrounding a that make all the points inside them have a value within the current range e, the same thing would hold if d is any smaller, so we can say "exist d > 0 such that ...", and because d is also decreasing, if at a certain d its true, it will also hold for d/2, so a-(d/2), a+(d/2) could be simpified to a-d, a+d (just scale down d until its equal to the previous d/2), and in range notation (a-d, a+d), and since we are now centering around a, we doesnt need to check for e range around both of the border anymore and instead just check the range on a, so for all x within range (a-d, a+d) the condition it must hold is that |g(x) - L| < e with L being the value of the "rate of change" and g(x) is the rate of change with respect to a (using normal slope), so if a number L exist such that all the point surrounding it with distance d has the rate of change difference to L fall within e, that is the rate of change of the function at point a
or "for any ϵ>0 , we can find a δ > 0 such that if 0
Calc 1 student this year, amazingly put together video thank you for the review🙏🙏
In your complex number section, you really shouldn't have said that -1^2 = 1^2; that's too fundamental mistake to let it slip by. It must read (-1)^2 = 1^2.
This is really making me wanna learn absolutely everything about the Fourier transform and I am so in for it!